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Blog - Petri net programming (part 1) (Rev #32)

This page is a blog article in progress, written by David Tanzer. To discuss this article while it’s being written, visit the Azimuth Forum.

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Introduction

Petri nets are a simple model of computation with a range of modelling applications that include chemical reaction networks, manufacturing processes, and population biology dynamics. They are graphs through which entities flow and have their types transformed. They also have far-reaching mathematical properties which are the subject of an extensive literature. See the network theory series here on Azimuth for a panoramic and well-diagrammed introduction to the theory and applications of Petri nets.

In this article, I will introduce Petri nets and give an expository presentation of a program to simulate them. I am hoping that this will be of interest to anyone who wants to see how scientific concepts can be formulated and actualized in computer software. Here the simulator program will be applied to a “toy model” of a chemical reaction network for the synthesis and dissociation of H2O molecules. The source code for this program should give the reader some “clay” to work with.

This material will prepare you for the further topic of stochastic Petri nets, where the sequencing of the Petri net is probabilistically controlled.

Definition of Petri nets

A Petri net is a diagram with two kinds of nodes: container nodes, called species (or “places”, “states”), which can hold zero or more tokens, and transition nodes, which are “wired” to some containers called its inputs and some called its outputs. A transition can have multiple input or output connections to a single container.

When a transition fires, it removes one token from each input container, and adds one token to each output container. When there are multiple inputs or outputs to the same container, then that many tokens get removed or added.

The total state of the Petri net is described by a labelling function which maps each container to the number of tokens it holds. A transition is enabled to fire in a labelling if there are enough tokens at its input containers. If no transitions are enabled, the it is halted.

The sequencing is non-deterministic, because in a given labelling there may be several transitions that are enabled. Dataflow arises whenever one transition sends tokens to a container that is read by another transition.

Petri nets represent entity conversion networks, which consist of entities of different species, along with reactions that convert input entities to output entities. Each entity is symbolized by a token in the net, and all the tokens for a species are grouped into an associated container. The conversion of entities is represented by the transitions that transform tokens.

Example 1: Disease processes

Here’s an example discussed earlier on Azimuth. It describes the virus that causes AIDS. The species are healthy cells, infected cells, and virion (technical term for an individual virus). The transitions are for infection, production of healthy cells, reproduction of virions within an infected cell, death of healthy cells, death of infected cells, and death of virions.

Here the species are yellow circles and the transitions are aqua squares. Note that there are three transitions called “death” and two called “production.” They are disambiguated by a Greek letter suffix.

production (α\alpha) describes the birth of one healthy cell, so it has no input and one healthy as output.

death (γ\gamma) has one healthy as input and no output.

death (δ\delta) has one infected as input and no output.

death (ζ\zeta) has one virion as input and no output.

infection (β\beta) takes one healthy and one virion as input, and has one infected cell as output.

production (ϵ\epsilon) describes the reproduction of the virus in an infected cell, so it has one infected as input, and one infected and one virion as output.

Example 2: Population dynamics

Here the tokens represent organisms, and the species are biological species. This simple example involving two species, wolf and rabbit:

There are three transitions: birth, which inputs one rabbit and outputs rabbit + rabbit like asexual reproduction), predation, which converts rabbit plus wolf to wolf plus wolf, and death, which inputs one wolf and outputs nothing.

Example 3: Chemical reaction networks

Here the entities are chemical units: molecules, isolated atoms, or ions. Each chemical unit is represented by a token in the net, and a container holds all of the tokens for a chemical species. Chemical reactions are then modeled as transitions that consume input tokens and generate output tokens.

We will be using the following simplified model for water formation and dissociation:

• The species are H, O, and H2O.

• Transition combine inputs two H atoms and one O atom, and outputs an H2O molecule.

• Transition split is the reverse process, which inputs one H2 and outputs two H and one O.

Note that the model is intended to show the idea of chemical reaction Petri nets, so it is not physically realistic. The actual reactions involve H2 and O2, and there are intermediate transitions as well. For more details, see Part 3 of the network theory series.

A Petri net simulator

The following Python script will simulate a Petri net, using parameters that describe the species, the transitions, and the initial labelling. It will run the net for a specified number of steps. At each step it chooses randomly among the enabled transitions, fires it, and prints the new labelling on the console.

Download

Here is the first Petri net simulator.

Running the script

The model parameters are already coded into the script. So let’s give it a whirl:

python petri1.py

This produced the output:

H, O, H2O, Transition
5, 3, 4, split
7, 4, 3, split
9, 5, 2, combine
7, 4, 3, combine
5, 3, 4, split
7, 4, 3, split
9, 5, 2, split
11, 6, 1, combine
9, 5, 2, split
11, 6, 1, split
13, 7, 0, ...

We started out in a state with 5 H’s, 3 O’s, and 2 H2O’s, then a split took place, which increased H by 2, increased O by 1, and decreased H2O by one, then…

Running it again gives a different execution sequence.

Top-level structure of the script

First construct the Petri net:

# combine: 2H + 1O -> 1H2O
combineSpec = ("combine", [["H",2],["O",1]], [["H2O",1]])

# split: 1H2O -> 2H + 1O 
splitSpec = ("split", [["H2O",1]], [["H",2],["O",1]])

petriNet = PetriNet(
    ["H","O","H2O"],         # species
    [combineSpec,splitSpec]  # transitions
)

Then establish the initial labelling:

initialLabelling = {"H":5, "O":3, "H2O":4}

Then run it for twenty steps:

steps = 20
petriNet.RunSimulation(steps, initialLabelling)

Program code

Species will be represented simply by their names, i.e., as strings. PetriNet and Transition will be defined as object classes. Each PetriNet instance holds a list of species names, a list of transition names, a list of Transition objects, and the current labelling. The labelling is a dictionary from species name to token count. Each Transition object contains an input dictionary and an input dictionary. The input dictionary map the name of a species to the number of times that the transition takes it as input, and similarly for the output dictionary.

Class PetriNet

The PetriNet class has a top-level method called RunSimulation, which makes repeated calls to FireOneRule. FireOneRule obtains the list of enabled transitions, chooses one randomly, and fires it. This is accomplished by the method SelectRandom, which uses a random integer between 1 and N to choose a transition from the list of enabled transitions.

class PetriNet(PetriNetBase):
    def RunSimulation(this, iterations, initialLabelling): 
        this.PrintHeader()  # prints e.g. "H, O, H2O"
        this.labelling = initialLabelling
        this.PrintLabelling() # prints e.g. "3, 5, 2" 

        for i in range(iterations):
           if this.IsHalted():
               print "halted"
               return 
           else:
               this.FireOneRule()
               this.PrintLabelling(); 
        print "iterations completed" 

    def EnabledTransitions(this):
        return filter(lambda transition: transition.IsEnabled(this.labelling), this.transitions)

    def IsHalted(this):
        return len(this.EnabledTransitions()) == 0

    def FireOneRule(this):
        this.SelectRandom(this.EnabledTransitions()).Fire(this.labelling)

    def SelectRandom(this, items):
        randomIndex = randrange(len(items))
        return items[randomIndex]
Class Transition

The Transition class exposes two key methods. IsEnabled takes a labeling as parameter, and returns a boolean saying whether the transition can fire. This is determined by comparing the input map for the transition with the token counts in the labeling, to see if there is sufficient tokens for it to fire. The Fire method takes a labelling in, and updates the counts in it to reflect the action of removing input tokens and creating output tokens.

class Transition:
    # Fields:
    # transitionName
    # inputMap: speciesName -> inputCount
    # outputMap: speciesName -> outputCount 

    # constructor
    def __init__(this, transitionName):
       this.transitionName = transitionName
       this.inputMap = {} 
       this.outputMap = {} 

    def IsEnabled(this, labelling):
       for inputSpecies in this.inputMap.keys():
          if labelling[inputSpecies] < this.inputMap[inputSpecies]: 
              return False  # not enough tokens
       return True # good to go 

    def Fire(this, labelling):
       print this.transitionName
       for inputName in this.inputMap.keys():
          labelling[inputName] = labelling[inputName] - this.inputMap[inputName] 
       for outputName in this.outputMap.keys():
          labelling[outputName] = labelling[outputName] + this.outputMap[outputName] 
Class PetriNetBase

Notice that the class line for PetriNet declares that it inherits from a base class PetriNetBase. The base class contains utility methods that support PetriNet: PrintHeader, PrintLabelling, SelectRandom, and the constructor, which converts the transition specifications into Transition objects.

class PetriNetBase:
    # Fields:
    # speciesNames
    # Transition list
    # labelling: speciesName -> token count 

    # constructor
    def __init__(this, speciesNames, transitionSpecs):
        this.speciesNames = speciesNames
        this.transitions = this.BuildTransitions(transitionSpecs)

    def BuildTransitions(this, transitionSpecs):
        transitions = []
        for (transitionName, inputSpecs, outputSpecs) in transitionSpecs:
            transition = Transition(transitionName)
            for degreeSpec in inputSpecs:
                this.SetDegree(transition.inputMap, degreeSpec)
            for degreeSpec in outputSpecs:
                this.SetDegree(transition.outputMap, degreeSpec)
            transitions.append(transition)
        return transitions 

    def SetDegree(this, dictionary, degreeSpec):
        speciesName = degreeSpec[0]
        if len(degreeSpec) == 2:
            degree = degreeSpec[1]
        else: 
            degree = 1
        dictionary[speciesName] = degree

    def PrintHeader(this):
        print string.join(this.speciesNames, ", ") + ", Transition"
           
    def PrintLabelling(this):
        for speciesName in this.speciesNames:
            print str(this.labelling[speciesName]) + ",", 

Summary

We’ve learned about an important model for computation, called Petri nets, and seen how it can be used to model a general class of entity conversion networks, which include chemical reaction networks as a major case. Then we wrote a program to simulate them, and applied it to a simple model for formation and dissociation of water molecules.

This is a good beginning, but observe the following limitation of our current program: it just randomly picks a rule. When we run the program, the simulation makes a kind of random walk, back and forth, between the states of full synthesis and full dissociation. But in a real chemical system, the rates at which the transitions fires are probabilistically determined, and depend, among other things, on the temperature. With a high probability for formation, and a low probability for dissociation, we would expect the system to reach an equilibrium state in which H2O is the predominant “token” in the system. The relative concentration of the various chemical species would be determined by the relative firing rates of the various transitions.

This gives motivation for the next article that I am writing, on stochastic Petri nets.

Programming exercises

  1. Extend the constructor for PetriNet to accept a dictionary from transitionName to a number, which will give the relative probability of that transition firing. Modify the firing rule logic to use these values. This is a step in the direction of the stochastic Petri nets covered in the next article.

  2. Make a prediction about the overall evolution of the system, given fixed probabilities for synthesis and dissociation, and then run the program to see if your prediction is confirmed.

  3. If you like, improve the usability of the script, by passing the model parameters from the command line. Use sys.argv and eval. You can use single quotes, and pass a string like “{‘H’:5, ‘O’:3, ‘H2O’:4}” from the command line.

By the way: if you do this exercises, please post a comment including your code, or a link to your code! To make Python code look pretty on this blog, see this.

Acknowledgements

Thanks to David Tweed and Ken Webb for reviewing the code and coming up with improvements to the specification syntax for the Petri nets. Thanks to Jacob Biamonte for the diagrams, and to John Baez for contributing the examples, and for editorial support.

Appendix: Notes on the language interpreter

The sample program is written in Python, which is a low-fuss scripting language with abstraction capabilities. The language is well-suited for proof-of-concept programming, and it has a medium-sized user base and support community. Here is the main website python.org.

You have a few distributions to choose from:

• If you are on a Linux or Mac type of system, it is likely to already be installed. Just open a shell and type “python.” Otherwise, use the package manager to install it.

• In Windows, you can use the version from the python.org web site. Alternatively, install cygwin, and in the installer, select Python, along with any of your other favorite Linux-style packages. Then you can partially pretend that you are working on a Linux type of system.

• The Enthought Python distribution comes packaged with a suite of Python-based open-source scientific and numeric computing packages. This includes ipython, scipy, numpy and matplotlib.

The program is distributed as a self-contained script, so in Linux and cygwin, you can just execute ./petri1.py. When running this way under cygwin, you can either refer to the cygwin version of the Python executable, or to any of the native Windows versions of interpreter. You just need to adjust the first line of the script to point to the python executable.

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